Related papers: Oriented Interval Greedoids
An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of $+1$ or $-1$. We define the adjacency, incidence and Laplacian matrices of an oriented hypergraph and study each of them. We extend several matrix…
The maximum intersection problem for a matroid and a greedoid, given by polynomial-time oracles, is shown $NP$-hard by expressing the satisfiability of boolean formulas in 3-conjunctive normal form as such an intersection. The corresponding…
Consider a collection of points in the plane and the sets of slopes or directions of the lines between pairs of points. It is known that the algebraic matroid on the set of direction constraints between the points is equivalent to the…
For an oriented matroid M, and given a generic single element extension and a generic single element lifting of M, the main result of [1] provides a bijection between bases of M and certain reorientations of M induced by the…
In 1980, Las Vergnas defined a notion of discrete convexity for oriented matroids, which Edelman subsequently related to the theory of anti-exchange closure functions and convex geometries. In this paper, we use generalized matroid activity…
We introduce a notion of duality (due to Brylawski) that generalizes matroid duality to arbitrary rank functions. This generalized duality allows for generalized operations (deletion and contraction) and a generalized polynomial based on…
We introduce a greedy algorithm optimizing arrangements of lines with respect to a property. We apply this algorithm to the case of simpliciality: it recovers all known simplicial arrangements of lines in a very short time and also produces…
In this series we introduce and investigate the concept of connectoids, which captures the connectivity structure of various discrete objects like undirected graphs, directed graphs, bidirected graphs, hypergraphs or finitary matroids. In…
We characterize which systems of sign vectors are the cocircuits of an oriented matroid in terms of the cocircuit graph.
In the paper we study the algebroid A of the groupoid of partially invertible elements over the lattice of orthogonal projections of a $W^*$-algebra. In particular the complex analytic manifold structure of these objects is investigated.…
A phased matroid is a matroid with additional structure which plays the same role for complex vector arrangements that oriented matroids play for real vector arrangements. The realization space of an oriented (resp., phased) matroid is the…
In his seminal 1983 paper, Jim Lawrence introduced lopsided sets and featured them as asymmetric counterparts of oriented matroids, both sharing the key property of strong elimination. Moreover, symmetry of faces holds in both structures as…
An oriented hypergraph is an oriented incidence structure that extends the concept of a signed graph. We introduce hypergraphic structures and techniques central to the extension of the circuit classification of signed graphs to oriented…
A matroid is a machine capturing linearity of mathematical objects and producing combinatorial structures. Matroid structure arises everywhere since linearity is a ubiquitous concept. One natural way to obtain matroids is by considering…
A transversal matroid whose dual is also transversal is called bi-transversal. Let $G$ be an undirected graph with vertex set $V$. In this paper, for every subset $W$ of $V$, we associate a bi-transversal matroid to the pair $(G,W)$. We…
We initiate the study of a class of polytopes, which we coin polypositroids, defined to be those polytopes that are simultaneously generalized permutohedra (or polymatroids) and alcoved polytopes. Whereas positroids are the matroids arising…
Develin and Sturmfels showed that regular triangulations of $\Delta_{n-1} \times \Delta_{d-1}$ can be thought as tropical polytopes. Tropical oriented matroids were defined by Ardila and Develin, and were conjectured to be in bijection with…
Zaslavsky (1991) introduced a graphical structure called a biased graph and used it to characterize all single-element coextensions and elementary lifts of graphic matroids. We introduce a new, dual graphical structure that we call a…
Threshold graphs are a prevalent and widely studied class of simple graphs. They have several equivalent definitions which makes them a go-to class for finding examples and counter examples when testing and learning. This versatility has…
We define braid presentation of edge-oriented spatial graphs as a natural generalization of braid presentation of oriented links. We show that every spatial graph has a braid presentation. For an oriented link it is known that the braid…